David Cameron is to signal his determination to maintain Britain’s membership of the EU by issuing a direct challenge to anti-EU campaigners who claim the UK could negotiate a semi-detached relationship with Brussels.

Amid growing fear among pro-EU Tories that the Leave campaign is making most of the running in the EU debate, Cameron will say he sees little future for Britain alongside European countries that have opted out of the EU.

The prime minister, who has faced strong pressure from pro-European Tories to highlight the benefits of the UK’s EU membership, will speak out during an appearance at the Northern Future Forum in Reykjavik, Iceland.

He will use the chance of a summit in a European capital outside the EU to issue a point-by-point challenge to the claim by the Leave campaign that Britain could negotiate a credible relationship with Brussels from outside the EU.

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Cameron will focus on the “Norwegian option”, often cited by the Leave campaign, which has allowed Oslo to enjoy access to the single market without joining the EU.

“The prime minister believes it is important to highlight the questions Britain would face if it left the EU and followed Norway’s model,” a No 10 source said. Downing Street pointed out that Norway is the 10th largest contributor to the EU budget and is bound by the rules of the single market without any say in the decision-making process.

Cameron’s intervention is a conscious decision to up the tempo on the EU ahead of the first formal roundtable negotiations on Britain’s demands at the European summit in December.

Downing Street had always decided it would adopt a low profile in the run-up to the Polish election last Sunday that resulted in the election of the Tories’ sister party, the Law and Justice party.

Cameron will now engage more actively on two levels. He will embark on a new round of meetings with EU leaders – he hosted Luxembourg’s prime minister, Xavier Bettel, in No 10 on Tuesday – before in mid-November laying down in writing his demands to the European council president, Donald Tusk. Cameron will also adopt a less neutral approach as he challenges the claims of the Leave campaign.

Downing Street sources said the new tactics do not mean the prime minister has decided that the UK should remain a member of the EU regardless of the outcome of his negotiations.

The No 10 source said: “The prime minister’s plan is to renegotiate the terms of the UK’s relationship with the EU – with the aim of staying in a reformed EU. But he rules nothing out if that negotiation is not successful.” That is code for saying that Cameron would be prepared to recommend a vote to leave the EU if his negotiations fail.

But senior Tories say there is a growing realisation in Downing Street that the prime minister would face a difficult finale to his premiership if the negotiations fail. George Osborne, who has long made clear in private that Britain would suffer a major loss of influence if it left the EU, would be severely damaged by a failure.

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Cameron, who will meet the Norwegian prime minister, Erna Solberg, and the Icelandic prime minister, Sigmundur Davíð Gunnlaugsson, in Reykjavik, will challenge the Leave campaign to answer five questions if the UK followed the Norwegian option:

Would the UK still have to follow EU rules, with no vote over how they are drawn up? A report by the Norwegian government says Oslo accepts about three quarters of EU rules with no votes in the EU institutions that set them.

Would the UK still be obliged to follow EU rules on free movement? Norway is a member of the European Economic Area, whose members are bound by free movement.

Would the UK still be forced to pay an EU subscription fee? Norway’s Europe minister said the country’s contributions are “more or less on par” with what it would pay if it were an EU member state.

What would happen to the UK’s existing trading deals? The UK would have to leave the EU’s common trade policy, which covers 52 states.

Would 31 other governments and parliaments agree to the UK’s new relationship with the EU? The deal would have to be agreed by the 27 other member states plus Iceland, Norway, Liechtenstein and Switzerland.

A Guardian article by the former Norwegian foreign minister, Espen Eide, offers help to Cameron. Eide writes: “Those campaigning for Britain to leave the EU and choose the Norwegian way can hence correctly claim that a country can retain access to the single market from outside the EU.

“What is normally not said, however, is that this also means retaining all the EU’s product standards, financial regulations, employment regulations, and substantial contributions to the EU budget. A Britain choosing this track would, in other words, keep paying, it would be ‘run by Brussels’, and it would remain committed to the four freedoms, including free movement.”

Speaking to the BBC, Frans Timmermans, vice-president of the European commission, confirmed that Norway – which is part of the Schengen open-borders agreement – would have to be part of the EU’s plan for resettling migrants.

He argued that Britain “is already in a position to control its own borders”, but indicated it would probably be possible to accommodate the UK’s demand for an exemption from “ever closer union” and greater powers to set domestic VAT rates.

The Vote Leave campaign director Dominic Cummings, who used to serve as Michael Gove’s special adviser, said: “Vote Leave does not support the ‘Norway option’ for Britain.

“After we vote leave, we will negotiate a new UK-EU deal based on free trade and friendly cooperation. We will end the supremacy of EU law.

“We will bring back control including over trade deals and migration. This is safer than voting to remain which means giving more power and money to Brussels every year.”